Explicit Modelling of Duration in HMM: an Efficient Algorithm

Abstract

Hidden Markov Modeling (HMM) techniques have been applied successfully to speech recognition problems. However, it has been claimed [1]-[5] that a major weakness of HMM is that the state duration probability density functions (SDPDF) are exponential, which is not appropriate for speech signals. In order to cope with this deficiency some authors have proposed to model explicitly the state duration. In these models the first order Markov hypothesis is broken in the loop transitions. Thus, the new models have been called Hidden Semi-Markov Models (HSMM). The first idea, up to the authors knowledge, is due to Fergurson [1] and consists in explicitly define a probability function per state, Pi, which controls the occupancy in each state. In his paper, Fergurson estimated Pi(d) from training data. One of the problems of this model is the large number of parameters per state (D, being D the maximum duration in any state). Those parameters have to be estimated in addition to those of the usual HMM. Therefore, an enormous database is required to accurately estimate the models. Fergurson himself suggested the possibility of using parametric functions for reducing the number of parameters, Levinson [3] extended the Baum Welch algorithm and provedits convergency. He also gave the details when the Gamma function is chosen as the PDF. Rusell and Moore [2] used the same result to recognize speech but by means of a Poisson function. Falachi [4] used a particular function chosen to increase the algorithm efficiency. Gu, Tseng and L. Lee [5] proposed the use of bounded functions (exponential functions lower and upper bounded) as a direct and simple (in training) but effective way of modelling the temporal structures existing in speech signals. In this paper an efficient algorithm to find the best state sequence in HSMM is presented. In next section we review the computational burden of these approximations and state a theorem which can effectively reduce their complexity. It is especially suitable to reduce complexity of HSMM as those proposed in [2, 3, 4].

This work has been supported by the grant TIC 92-1026-C02/02